Legendrian contact instanton cohomology and its spectral invariants on the one-jet bundle

In the present paper, we develop the Floer-style elliptic Morse theory for the Hamiltonian-perturbed contact action functional attached to the Legendrian links. Motivated by the present authors' construction [OY2] of the a perturbed action functional defined on the Carnot path space introduced in [OY2] as the canonical generating function, we apply a Floer-type theory to the aforementioned functional and associate the Legendrian contact instanton cohomology, denote by $HI^*(J^1B,H;R)$, to each Legendrian submanifold contact isotopic to the zero section of one-jet bundle. Then we give a Floer theoretic construction of Legendrian spectral invariants and establish their basic properties. This theory subsumes the Lagrangian intersection theory and spectral invariants on the cotangent bundle previously developed by the first-named author in [Oh1,Oh2]. The main ingredient for the study is the interplay between the geometric analysis of the Hamiltonian-perturbed contact instantons and the calculus of contact Hamiltonian geometry.